Probabilistic Reasoning With Bayes Rule Outline Motivation Generalizing
Probabilistic Reasoning With Bayes’ Rule Outline: Motivation. Generalizing Modus Ponens Bayes’ Rule Applying Bayes’ Rule Odds-Likelihood Formulation of Bayes’ Rule. CSE 415 -- (c) S. Tanimoto, 2004 Probabilistic Reasoning 1
Motivation Logical reasoning has limitations: It requires that assumptions be considered “certain”. It typically uses general rules. General rules that are reliable may be difficult to come by. Logical reasoning can be awkward for certain structured domains such as time and space. CSE 415 -- (c) S. Tanimoto, 2004 Probabilistic Reasoning 2
Generalizing Modus Ponens: P -> Q P -----Q Bayes’ Rule: (general idea) If P then sometimes Q P ---------------Maybe Q (Bayes’ rule lets us calculate the probability of Q, taking P into account. ) CSE 415 -- (c) S. Tanimoto, 2004 Probabilistic Reasoning 3
Bayes’ Rule E: Some evidence exists, i. e. , a particular condition is true H: some hypothesis is true. P(E|H) = probability of E given H. P(E|~H) = probability of E given not H. P(H) = probability of H, independent of E. P(E|H) P(H|E) = --------P(E) = P(E|H) P(H) + P(E|~H)(1 - P(H)) CSE 415 -- (c) S. Tanimoto, 2004 Probabilistic Reasoning 4
Applying Bayes’ Rule E: The patient’s white blood cell count exceeds 110% of average. H: The patient is infected with tetanus. P(E|H) = 0. 8 P(E|~H) = 0. 3 P(H) = 0. 01 class-conditional probability “ prior probability posterior probability: P(E|H) P(H|E) = --------P(E) (0. 8) (0. 01) 0. 008 = ---------------- = 0. 0262 (0. 8) (0. 01) + (0. 3)(0. 99) 0. 305 P(E) = P(E|H) P(H) + P(E|~H)(1 - P(H)) CSE 415 -- (c) S. Tanimoto, 2004 Probabilistic Reasoning 5
Odds are 10 to 1 it will rain tomorrow. 10 10 P(rain) = ------10 + 1 11 Suppose P(A) = 1/4 Then O(A) = (1/4) / (3/4) = 1/3 in general: P(A) O(A) = -------P(~A) CSE 415 -- (c) S. Tanimoto, 2004 Probabilistic Reasoning P(A) = -----1 - P(A) 6
Bayes’ Rule reformulated. . . P(H|E) = _______ P(E|H) P(H) --------P(E) _______ P(~H|E) = P(E|~H) P(~H) ----------P(E) O(H|E) = P(E|H) ------ O(H) P(E|~H) CSE 415 -- (c) S. Tanimoto, 2004 Probabilistic Reasoning 7
Odds-Likelihood Form of Bayes’ Rule E: The patient’s white blood cell count exceeds 110% of average. H: The patient is infected with tetanus. O(H) = 0. 01/0. 99 = λ O(H) lambda is called the sufficiency factor. O(H|~E) = λ’ O(H) lambda prime is called the necessity factor. O(H|E) CSE 415 -- (c) S. Tanimoto, 2004 Probabilistic Reasoning 8
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